Theorems · Theorem · functional analysis

Orthonormal.inner_left_finsupp

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : SeminormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_4} {v : ι → E},
  Orthonormal 𝕜 v → ∀ (l : ι →₀ 𝕜) (i : ι), inner 𝕜 ((Finsupp.linearCombination 𝕜 v) l) (v i) = (starRingEnd 𝕜) (l i)

The inner product of a linear combination of a set of orthonormal vectors with one of those vectors picks out the coefficient of that vector.

Defined in

Mathlib.Analysis.InnerProductSpace.Orthonormal

Cited by

2 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeSeminormedAddCommGroupInnerProductSpace